# Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique

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## Abstract

**:**

## 1. Introduction

## 2. General Observations and Notions

#### 2.1. Some Concepts on the COVID-19 Model

- If there is no COVID-19, the model ${\xi}_{0}=\left({S}_{*}^{0},{E}_{*}^{0},{I}_{*}^{0},{Q}_{*}^{0},{R}_{*}^{0}\right)=(\delta /\psi ,0,0,0,0)$ is an equilibrium point.
- In general, the equilibrium takes the form ${\xi}_{1}=\left({S}_{*}^{1},{E}_{*}^{1},{I}_{*}^{1},{Q}_{*}^{1},{R}_{*}^{1}\right)$, where$${S}_{*}^{1}=\frac{\left({q}_{1}+\xi +\gamma +\psi \right)\left(K+\psi +{\nu}_{1}\right)}{{\beta}_{1}\gamma +{\beta}_{2}\left(K+\psi +{\nu}_{1}\right)},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{E}_{*}^{1}=\frac{\left(\delta -\psi {S}_{*}^{1}\right)\left(K+\psi +{\nu}_{1}\right)}{\left[{\beta}_{1}\gamma +{\beta}_{2}\left(K+\psi +{\nu}_{1}\right)\right]{S}_{*}^{1}},$$$${I}_{*}^{1}=\frac{\gamma {E}_{*}^{1}}{K+\psi +{\nu}_{1}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{Q}_{*}^{1}=\frac{{q}_{1}{E}_{*}^{1}}{\chi +\psi +{\nu}_{2}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{R}_{*}^{1}=\frac{\xi {E}_{*}^{1}+K{I}_{*}^{1}+\chi {Q}_{*}^{1}}{\psi}.$$

#### 2.2. Non-Negative Solutions, Equilibrium Points and Stability

**Theorem**

**1**

- 1.
- ${(S\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right),R\left(t\right))}^{T}$ is the unique solution of (1) and leftover in ${R}_{+}^{5}$.
- 2.
- If ${\mathfrak{R}}_{0}<1$, then the disease free equilibrium point ${\xi}_{0}=({S}_{*}^{0},\phantom{\rule{0.166667em}{0ex}}{E}_{*}^{0},\phantom{\rule{0.166667em}{0ex}}{I}_{*}^{0},\phantom{\rule{0.166667em}{0ex}}{Q}_{*}^{0},\phantom{\rule{0.166667em}{0ex}}{R}_{*}^{0})$ is locally asymptotically stable (LAS).
- 3.
- The endemic equilibrium point ${\xi}_{1}=({S}_{*}^{1},\phantom{\rule{0.166667em}{0ex}}{E}_{*}^{1},\phantom{\rule{0.166667em}{0ex}}{I}_{*}^{1},\phantom{\rule{0.166667em}{0ex}}{Q}_{*}^{1},\phantom{\rule{0.166667em}{0ex}}{R}_{*}^{1})$ is LAS iff ${\mathfrak{R}}_{0}>1$.

## 3. Numerical Implementation of the MSRM

## 4. Error Analysis

**Theorem**

**2.**

**Proof.**

**Theorem**

**3**

## 5. Numerical Simulation

- Figure 2 shows how the approximate solution behaves for various approximation orders, $m=4$ (Figure 2a) and $m=6$ (Figure 2b), in the domain ($0,25$) with ${S}_{0}=0.5,\phantom{\rule{0.166667em}{0ex}}{E}_{0}=0.2,\phantom{\rule{0.166667em}{0ex}}{I}_{0}={Q}_{0}={R}_{0}=0.1$ and the parameters $\delta =0.5,\phantom{\rule{0.166667em}{0ex}}{\beta}_{1}=1.05,\phantom{\rule{0.166667em}{0ex}}{\beta}_{2}=0.005,\phantom{\rule{0.166667em}{0ex}}\psi =0.5,\phantom{\rule{0.166667em}{0ex}}{q}_{1}=0.001,\phantom{\rule{0.166667em}{0ex}}\xi =0.00398,\phantom{\rule{0.166667em}{0ex}}{\nu}_{1}=0.0047876,\phantom{\rule{0.166667em}{0ex}}{\nu}_{2}=0.000001231,\phantom{\rule{0.166667em}{0ex}}\gamma =0.085432,\phantom{\rule{0.166667em}{0ex}}K=0.09871,\phantom{\rule{0.166667em}{0ex}}\chi =0.1243.$ In this case, ${\mathfrak{R}}_{0}=0.130112<1$, and Theorem 1 also allows us to observe that the disease-free equilibrium point ${\xi}_{0}$ is LAS.
- Figure 3 gives the behavior of the approximate solution under the distinct initial solution with $m=6$ in the interval ($0,75$), and the parameters $\delta =0.5,\phantom{\rule{0.166667em}{0ex}}{\beta}_{1}=1.05,\phantom{\rule{0.166667em}{0ex}}{\beta}_{2}=0.005,\phantom{\rule{0.166667em}{0ex}}\psi =0.5,\phantom{\rule{0.166667em}{0ex}}{q}_{1}=0.001,\phantom{\rule{0.166667em}{0ex}}\xi =0.00398,\phantom{\rule{0.166667em}{0ex}}{\nu}_{1}=0.0047876,\phantom{\rule{0.166667em}{0ex}}{\nu}_{2}=0.000001231,\phantom{\rule{0.166667em}{0ex}}\gamma =0.085432,\phantom{\rule{0.166667em}{0ex}}K=0.09871,\phantom{\rule{0.166667em}{0ex}}\chi =0.1243$; where $S\left(t\right),\phantom{\rule{0.166667em}{0ex}}E\left(t\right),\phantom{\rule{0.166667em}{0ex}}I\left(t\right),\phantom{\rule{0.166667em}{0ex}}Q\left(t\right),\phantom{\rule{0.166667em}{0ex}}R\left(t\right)$ are plotted in Figure 3a–e, respectively. Here, we give the following three cases:
**i**.- ${S}_{0}=0.5,\phantom{\rule{0.166667em}{0ex}}{E}_{0}=0.2,\phantom{\rule{0.166667em}{0ex}}{I}_{0}={Q}_{0}={R}_{0}=0.1$;
**ii**.- ${S}_{0}=0.6,\phantom{\rule{0.166667em}{0ex}}{E}_{0}=0.3,\phantom{\rule{0.166667em}{0ex}}{I}_{0}={Q}_{0}={R}_{0}=0.2$;
**iii**.- ${S}_{0}=0.7,\phantom{\rule{0.166667em}{0ex}}{E}_{0}=0.4,\phantom{\rule{0.166667em}{0ex}}{I}_{0}={Q}_{0}={R}_{0}=0.3$.

In the above three cases, we can confirm that the condition of stability is satisfied, i.e., ${\mathfrak{R}}_{0}<1$. - The approximate solutions provided by the MSRM and RK4 methods (at $h=0.05$) are compared in Figure 6 with the same parameters and initial conditions as in Figure 2. This graphic demonstrates that the theoretical stability findings found in the previous section are accurate.As shown in Figure 2 through Figure 6, the approximate solution is affected by the values of $m,\phantom{\rule{0.166667em}{0ex}}{q}_{1}$, the initial conditions, and the added parameters $\delta ,\phantom{\rule{0.166667em}{0ex}}{\beta}_{1},\phantom{\rule{0.166667em}{0ex}}{\beta}_{2},\phantom{\rule{0.166667em}{0ex}}\psi ,\phantom{\rule{0.166667em}{0ex}}\xi ,\phantom{\rule{0.166667em}{0ex}}{\nu}_{1},$$\phantom{\rule{0.166667em}{0ex}}{\nu}_{2},\phantom{\rule{0.166667em}{0ex}}\gamma ,\phantom{\rule{0.166667em}{0ex}}K,\phantom{\rule{0.166667em}{0ex}}\chi $. This shows that the suggested technique can be effectively applied to solve the presented model.

## 6. Conclusions

- The suggested approach is efficient and reliable.
- The approach has the capacity to apply a limited number of series solution terms to produce precise results.
- There are several benefits to using this approach to solve this kind of problem.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Adel, M.; Khader, M.M.; Assiri, T.A.; Kallel, W.
Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique. *Symmetry* **2023**, *15*, 931.
https://doi.org/10.3390/sym15040931

**AMA Style**

Adel M, Khader MM, Assiri TA, Kallel W.
Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique. *Symmetry*. 2023; 15(4):931.
https://doi.org/10.3390/sym15040931

**Chicago/Turabian Style**

Adel, Mohamed, Mohamed M. Khader, Taghreed A. Assiri, and Wajdi Kallel.
2023. "Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique" *Symmetry* 15, no. 4: 931.
https://doi.org/10.3390/sym15040931